The number of digits in `(48^(4)xx5^(12))` is

Options

A. 18
B. 16
C. 14
D. 12

Answer

To find the number of digits in the expression \( 48^4 \times 5^{12} \), we can follow these steps:

### Step 1: Simplify the expression
We start with the expression:
\[ 48^4 \times 5^{12} \]
We can break down \( 48 \) into its prime factors:
\[ 48 = 3 \times 16 = 3 \times 2^4 \]
Thus,
\[ 48^4 = (3 \times 2^4)^4 = 3^4 \times (2^4)^4 = 3^4 \times 2^{16} \]
Now, substituting this back into the original expression:
\[ 48^4 \times 5^{12} = 3^4 \times 2^{16} \times 5^{12} \]

### Step 2: Combine the powers of 2 and 5
Next, we can combine the powers of \( 2 \) and \( 5 \):
\[ = 3^4 \times 2^{16} \times 5^{12} = 3^4 \times 2^{16} \times 5^{12} = 3^4 \times 2^{16} \times 5^{12} \]
We can factor out \( 2^{12} \) from \( 2^{16} \):
\[ = 3^4 \times 2^{12} \times 2^4 \times 5^{12} \]
Now, we can combine \( 2^{12} \) and \( 5^{12} \):
\[ = 3^4 \times (2 \times 5)^{12} \times 2^4 = 3^4 \times 10^{12} \times 2^4 \]

### Step 3: Calculate \( 3^4 \) and \( 2^4 \)
Now, we can calculate \( 3^4 \) and \( 2^4 \):
\[ 3^4 = 81 \quad \text{and} \quad 2^4 = 16 \]
Thus, we have:
\[ = 81 \times 10^{12} \times 16 \]

### Step 4: Calculate the product
Now, we can calculate \( 81 \times 16 \):
\[ 81 \times 16 = 1296 \]
So, we can rewrite our expression as:
\[ = 1296 \times 10^{12} \]

### Step 5: Determine the number of digits
To find the number of digits in \( 1296 \times 10^{12} \):
1. The number \( 1296 \) has \( 4 \) digits.
2. The \( 10^{12} \) contributes \( 12 \) additional digits.

Therefore, the total number of digits is:
\[ 4 + 12 = 16 \]

### Final Answer
The number of digits in \( 48^4 \times 5^{12} \) is \( 16 \).

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